Why Does the Multidimensional Chain Rule Lead to Euler's Theorem Proof?

[Lambda96]

Homework Statement

Proof $\sum\limits_{i=1}^{n} x_i \frac{\partial f}{\partial x_i}(x)=n f(x)$

Relevant Equations

Chain rule

Hi,

I am having problems with the following task:

My lecturer gave me the tip that I should apply the multidimensional chain rule to obtain the following transformation $\sum\limits_{i=1}^{n} \frac{\partial f}{\partial x_i}(x) \cdot x_i= \frac{d f(tx)}{dt} \big|_{t=1}$

Unfortunately, I don't know why the two terms should be equal because of the chain rule. There is a sum on the left and a derivative on the right side.

One more thing, in the 1D case the chain rule is $[f(g(x))]'=f'(g(x)) \cdot g'(x)$ but the term $g'(x)$ for the multidimensional case is missing on the right-hand side.

 

[pasmith]

You have two ways of calculating $\frac{d}{dt}f(tx)$. One is to apply the chain rule, as you would for any function: $$\frac{d}{dt}f(tx) = \sum_i \frac{\partial f}{\partial x_i} \frac{d}{dt}(tx_i).$$ The other is to use the particular feature of $f$ that $f(tx) = t^nf(x)$: $$\frac{d}{dt}f(tx) = \left(\frac{d}{dt} t^n\right)f(x).$$ These two expressions must be equal to each other. What then happens if you evaluate the derivatives and set $t = 1$?

 

[Lambda96]

Thank you pasmith for your help


I think I'm beginning to understand the derivation now. I was also wondering all the time how the $t$ gets into the equation, but since you evaluate the derivative for $t=1$, you can claim that the two derivatives are equal.